Blow-up profile of Bose-Einstein condensate with singular potentials

TL;DR

This paper investigates the collapse behavior of Bose-Einstein condensates governed by the attractive Gross-Pitaevskii equation with unbounded potentials, identifying the precise nature of the singularity formation as interaction strength reaches a critical level.

Contribution

It provides a detailed analysis of the blow-up profile and collapse dynamics of the condensate near the critical interaction strength in the presence of singular potentials.

Findings

Collapse occurs at a single singular point.

The blow-up profile is characterized precisely up to leading order.

The critical interaction strength for collapse is identified.

Abstract

The paper is concerned with the Bose-Einstein condensate described by the attractive Gross-Pitaevskii equation in R 2 , where the external potential is unbounded from below. We show that when the interaction strength increases to a critical value, the Gross-Pitaevskii minimizer collapses to one singular point and we analyze the details of the collapse exactly up to the leading order.